2,492 research outputs found
Optimal simplices and codes in projective spaces
We find many tight codes in compact spaces, i.e., optimal codes whose
optimality follows from linear programming bounds. In particular, we show the
existence (and abundance) of several hitherto unknown families of simplices in
quaternionic projective spaces and the octonionic projective plane. The most
noteworthy cases are 15-point simplices in HP^2 and 27-point simplices in OP^2,
both of which are the largest simplices and the smallest 2-designs possible in
their respective spaces. These codes are all universally optimal, by a theorem
of Cohn and Kumar. We also show the existence of several positive-dimensional
families of simplices in the Grassmannians of subspaces of R^n with n <= 8;
close numerical approximations to these families had been found by Conway,
Hardin, and Sloane, but no proof of existence was known. Our existence proofs
are computer-assisted, and the main tool is a variant of the Newton-Kantorovich
theorem. This effective implicit function theorem shows, in favorable
conditions, that every approximate solution to a set of polynomial equations
has a nearby exact solution. Finally, we also exhibit a few explicit codes,
including a configuration of 39 points in OP^2 that form a maximal system of
mutually unbiased bases. This is the last tight code in OP^2 whose existence
had been previously conjectured but not resolved.Comment: 53 page
An Upper Bound On the Size of Locally Recoverable Codes
In a {\em locally recoverable} or {\em repairable} code, any symbol of a
codeword can be recovered by reading only a small (constant) number of other
symbols. The notion of local recoverability is important in the area of
distributed storage where a most frequent error-event is a single storage node
failure (erasure). A common objective is to repair the node by downloading data
from as few other storage node as possible. In this paper, we bound the minimum
distance of a code in terms of its length, size and locality. Unlike previous
bounds, our bound follows from a significantly simple analysis and depends on
the size of the alphabet being used. It turns out that the binary Simplex codes
satisfy our bound with equality; hence the Simplex codes are the first example
of a optimal binary locally repairable code family. We also provide
achievability results based on random coding and concatenated codes that are
numerically verified to be close to our bounds.Comment: A shorter version has appeared in IEEE NetCod, 201
New Quasi-Cyclic Codes from Simplex Codes
As a generalization of cyclic codes, quasi-cyclic (QC) codes contain many
good linear codes. But quasi-cyclic codes studied so far are mainly limited to
one generator (1-generator) QC codes. In this correspondence, 2-generator and
3-generator QC codes are studied, and many good, new QC codes are constructed
from simplex codes. Some new binary QC codes or related codes, that improve the
bounds on maximum minimum distance for binary linear codes are constructed.
They are 5-generator QC [93, 17, 34] and [254, 23, 102] codes, and related [96,
17, 36], [256, 23, 104] codes.Comment: 3 page
Bounds and Constructions of Locally Repairable Codes: Parity-check Matrix Approach
A -ary locally repairable code (LRC) is an linear code
over such that every code symbol can be recovered by accessing
at most other code symbols. The well-known Singleton-like bound says that
and an LRC is said to be optimal if it attains
this bound. In this paper, we study the bounds and constructions of LRCs from
the view of parity-check matrices. Firstly, a simple and unified framework
based on parity-check matrix to analyze the bounds of LRCs is proposed. Several
useful structural properties on -ary optimal LRCs are obtained. We derive an
upper bound on the minimum distance of -ary optimal -LRCs in terms
of the field size . Then, we focus on constructions of optimal LRCs over
binary field. It is proved that there are only 5 classes of possible parameters
with which optimal binary -LRCs exist. Moreover, by employing the
proposed parity-check matrix approach, we completely enumerate all these 5
classes of possible optimal binary LRCs attaining the Singleton-like bound in
the sense of equivalence of linear codes.Comment: 18 page
Kirkman Equiangular Tight Frames and Codes
An equiangular tight frame (ETF) is a set of unit vectors in a Euclidean
space whose coherence is as small as possible, equaling the Welch bound. Also
known as Welch-bound-equality sequences, such frames arise in various
applications, such as waveform design and compressed sensing. At the moment,
there are only two known flexible methods for constructing ETFs: harmonic ETFs
are formed by carefully extracting rows from a discrete Fourier transform;
Steiner ETFs arise from a tensor-like combination of a combinatorial design and
a regular simplex. These two classes seem very different: the vectors in
harmonic ETFs have constant amplitude, whereas Steiner ETFs are extremely
sparse. We show that they are actually intimately connected: a large class of
Steiner ETFs can be unitarily transformed into constant-amplitude frames,
dubbed Kirkman ETFs. Moreover, we show that an important class of harmonic ETFs
is a subset of an important class of Kirkman ETFs. This connection informs the
discussion of both types of frames: some Steiner ETFs can be transformed into
constant-amplitude waveforms making them more useful in waveform design; some
harmonic ETFs have low spark, making them less desirable for compressed
sensing. We conclude by showing that real-valued constant-amplitude ETFs are
equivalent to binary codes that achieve the Grey-Rankin bound, and then
construct such codes using Kirkman ETFs
New Construction of 2-Generator Quasi-Twisted Codes
Quasi-twisted (QT) codes are a generalization of quasi-cyclic (QC) codes.
Based on consta-cyclic simplex codes, a new explicit construction of a family
of 2-generator quasi-twisted (QT) two-weight codes is presented. It is also
shown that many codes in the family meet the Griesmer bound and therefore are
length-optimal. New distance-optimal binary QC [195, 8, 96], [210, 8, 104] and
[240, 8, 120] codes, and good ternary QC [208, 6, 135] and [221, 6, 144] codes
are also obtained by the construction.Comment: 4 page
Packing Lines, Planes, etc.: Packings in Grassmannian Space
This paper addresses the question: how should N n-dimensional subspaces of
m-dimensional Euclidean space be arranged so that they are as far apart as
possible? The results of extensive computations for modest values of N, n, m
are described, as well as a reformulation of the problem that was suggested by
these computations. The reformulation gives a way to describe n-dimensional
subspaces of m-space as points on a sphere in dimension (m-1)(m+2)/2, which
provides a (usually) lower-dimensional representation than the Pluecker
embedding, and leads to a proof that many of the new packings are optimal. The
results have applications to the graphical display of multi-dimensional data
via Asimov's "Grand Tour" method.Comment: 36 pages, 15 figure
New Construction of A Family of Quasi-Twisted Two-Weight Codes
Based on cyclic and consta-cyclic simplex codes, a new explicit construction
of a family of two-weight codes is presented. These two-weight codes obtained
are in the form of 2-generator quasi-cyclic, or quasi-twisted structure. Based
on this construction, new optimal binary quasi-cyclic [195, 8, 96], [210, 8,
104] and [240, 8, 120] codes, and good QC ternary [208, 6, 135] and [221, 6,
144] codes are thus obtained. It is also shown that many codes among the family
meet the Griesmer bound and thereful are optimal.Comment: 4 pages, submitted to IEEE Trans. Information Theor
Numerical cubature using error-correcting codes
We present a construction for improving numerical cubature formulas with
equal weights and a convolution structure, in particular equal-weight product
formulas, using linear error-correcting codes. The construction is most
effective in low degree with extended BCH codes. Using it, we obtain several
sequences of explicit, positive, interior cubature formulas with good
asymptotics for each fixed degree as the dimension . Using a
special quadrature formula for the interval [arXiv:math.PR/0408360], we obtain
an equal-weight -cubature formula on the -cube with O(n^{\floor{t/2}})
points, which is within a constant of the Stroud lower bound. We also obtain
-cubature formulas on the -sphere, -ball, and Gaussian with
points when is odd. When is spherically symmetric and
, we obtain points. For each , we also obtain explicit,
positive, interior formulas for the -simplex with points; for
, we obtain O(n) points. These constructions asymptotically improve the
non-constructive Tchakaloff bound.
Some related results were recently found independently by Victoir, who also
noted that the basic construction more directly uses orthogonal arrays.Comment: Dedicated to Wlodzimierz and Krystyna Kuperberg on the occasion of
their 40th anniversary. This version has a major improvement for the n-cub
Reconstruction Codes for DNA Sequences with Uniform Tandem-Duplication Errors
DNA as a data storage medium has several advantages, including far greater
data density compared to electronic media. We propose that schemes for data
storage in the DNA of living organisms may benefit from studying the
reconstruction problem, which is applicable whenever multiple reads of noisy
data are available. This strategy is uniquely suited to the medium, which
inherently replicates stored data in multiple distinct ways, caused by
mutations. We consider noise introduced solely by uniform tandem-duplication,
and utilize the relation to constant-weight integer codes in the Manhattan
metric. By bounding the intersection of the cross-polytope with hyperplanes, we
prove the existence of reconstruction codes with greater capacity than known
error-correcting codes, which we can determine analytically for any set of
parameters.Comment: 11 pages, 2 figures, Latex; version accepted for publicatio
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